Computability of the Hahn-Banach Theorem Revisited

2026-03-17

AI Summary

This study investigates the computability and Weihrauch complexity of the Hahn–Banach theorem restricted to the fixed Banach space ℓ¹. Employing methods from computable analysis and reverse mathematics, together with constructions involving infinite products and reduction techniques, the authors demonstrate that the theorem attains its full Weihrauch strength already in this setting: its general form is Weihrauch-equivalent to Weak Kőnig’s Lemma, the one-dimensional case is equivalent to the Intermediate Value Theorem, and the two-dimensional case corresponds to LLPO. These results establish novel equivalences between the Hahn–Banach theorem and these foundational principles, while also providing streamlined reductions that deepen our understanding of the computational content inherent in this central result of functional analysis.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Learning Theory

Application Category

• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions

Abstract

Computational properties of the Hahn-Banach theorem have been studied in computable, constructive and reverse mathematics and in all these approaches the theorem is equivalent to weak Kőnig's lemma. Gherardi and Marcone proved that this is also true in the uniform sense of Weihrauch complexity. However, their result requires the underlying space to be variable. We prove that the Hahn-Banach theorem attains its full complexity already for the Banach space $\ell^1$. We also prove that the one-step Hahn-Banach theorem for this space is Weihrauch equivalent to the intermediate value theorem. This also yields a new and very simple proof of the reduction of the Hahn-Banach theorem to weak Kőnig's lemma using infinite products. Finally, we show that the Hahn-Banach theorem for $\ell^1$ in the two-dimensional case is Weihrauch equivalent to the lesser limited principle of omniscience.

Problem

• Hahn-Banach theorem
• Weihrauch complexity
• computable analysis
• weak Kőnig's lemma
• Banach space

Innovation

• Hahn-Banach theorem
• Weihrauch complexity
• ℓ¹ space
• weak Kőnig's lemma
• intermediate value theorem